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1 MATH 321 EQUIVALENCE RELATIONS, WELL-DEFINEDNESS, MODULAR ARITHMETIC, AND THE RATIONAL NUMBERS ALLAN YASHINSKI Abstract. We explore the notion of well-definedness when defining functions whose domain is the set of all equivalence classes of an equivalence relation. Then we apply this to define modular arithmetic and the set Q of rational numbers. 1. Equivalence Classes We shall slightly adapt our notation for relations in this document. Let be a relation on a set X. Formally, is a subset of X X. Given two elements x, y X, we shall write x y to mean (x, y). From now on, we shall just use the notation x y, and not explicitly reference as a subset of X X. Recall that a relation on a set X is an equivalence relation if is (i) reflexive: ( x X)(x x), (ii) symmetric: ( x, y X)(x y y x), and (iii) transitive: ( x, y, z X) ( (x y y z) (x z) ). Suppose is an equivalence relation on X. When two elements are related via, it is common usage of language to say they are equivalent. Given x X, the equivalence class of x is the set x = {y X : x y}. In other words, the equivalence class x of x is the set of all elements of X that are equivalent to x. Be mindful that x is a subset of X, it is not an element of X. Typically, the set x contains much more than just x. The element x is called a representative of the equivalence class x. Any element of an equivalence class can serve as a representative. Exercise A. Given an equivalence relation on a set X, show that for any two elements x, y X, x y if and only if x = y. We shall use the notation X/ to mean the set of all equivalence classes with respect to. That is, X/ = { x : x X }. Notice that X/ is a set whose elements are subsets of X. We ve often referred to such a thing as a family of subsets of X. As shown in Theorem (i), X/ is actually a partition of X. 1

2 2 ALLAN YASHINSKI Example 1. Fix n N. Recall that we say two integers a, b Z are congruent modulo n when n (a b). As discussed in class, conguence modulo n is an equivalence relation. We shall write a b mod n to mean a is conguent to b modulo n. The set of equivalence classes of integers with respect to this equivalence relation is traditionally denoted Z/nZ. This is less cumbersome notationally than writing something like Z/( mod n). Exercise B. Consider the relation of congruence modulo 5. Explicitly describe the equivalence classes 0 and 7 from Z/5Z. 2. Functions whose domain is X/ It is common in mathematics (more common than you might guess) to work with the set X/ of equivalence classes of an equivalence relation. Issues arise when one attempts to define functions f : X/ Y whose domain is X/. When defining any function, one usually describes what the function does to a typical element of the domain. In this case, a typical element of the domain is an equivalence class x, which is represented by some element x X. One typically wants to define what f(x) is in terms of the representative x. This is where one can get into trouble. Consider the following example. Example 2. Let us attempt to define a function f : Z/5Z Z by the formula f(m) = m. Then we have f(0) = 0, f(2) = 2, f(3) = 3, f(7) = 7, and so on. But there is a major problem here. Since 2 7 mod 5, it follows that 2 = 7. That is, 2 and 7 are the exact same element of the set Z/5Z. However, the function we ve defined above has the property that f(2) f(7). This is a clear violation of the definition of a function, and so f is not really a function at all. Mathematicians would say that the function f is not well-defined, which really just means that f is not a function. The issue is that the rule for f maps different representatives of the same equivalence class to different elements of the codomain. Example 3. Let us attempt to define another function with domain Z/5Z. Given an integer m Z, let r 5 (m) denote the remainder, as in the Division Algorithm, after m is divided by 5. So r 5 (m) {0, 1, 2, 3, 4}. For example, Let s define a function r 5 (13) = 3, because 13 = f : Z/5Z {0, 1, 2, 3, 4}, by f(m) = r 5 (m). In this case, this function is well-defined (we ll explain how one checks this below.) At first glance, the definition of f(m) appears to depend on the choice of the representative m of the equivalence class, but it actually does not.

3 EQUIVALENCE RELATIONS AND WELL-DEFINEDNESS 3 We shall now explore how to ensure that we are defining actual functions out of X/. In what follows, let π : X X/ be the function defined by π(x) = x. Notice this is an honest function. It is surjective, but in general is not injective, because any two equivalent elements will have the same image. Exercise C. Suppose that f : X/ Y is a function, that is an honest welldefined function. Let g : X Y be the function g = f π. Prove that g(x) = g(x ) whenever x x. The point of this exercise is that in order to have a well-defined function f : X/ Y, then it must be the case that f(x) = f(x ) whenever x x. If not, then our attempt at making a function is not really a function at all. The following theorem says that this is the only issue we need to confront. Theorem 1. Let be an equivalence relation on X and let g : X Y be a function with the property that g(x) = g(x ) whenever x x. Then there exists a (well-defined) function given by the formula f(x) = g(x). f : X/ Y Before we prove this theorem, let us return to Example 3. We started with the function r 5 : Z {0, 1, 2, 3, 4}. Here, r 5 is going to play the role of g in the statement of the theorem. To use the theorem to verify that f(m) = r 5 (m) is well-defined, we must show that Assume m m r 5 (m) = r 5 (m ) whenever m m mod 5. mod 5. Then there is some k Z such that m = m + 5k. Applying the Division Algorithm to m gives us integers q and r such that m = 5q + r. Now r {0, 1, 2, 3, 4} and r 5 (m ) = r by definition of r 5. Combining the above equations shows that m = m + 5k = (5q + r) + 5k = 5(q + k) + r. From this, it follows that r 5 (m) = r. Thus we have shown that By the above theorem, the function is well-defined. r 5 (m) = r 5 (m ) whenever m m mod 5. f : Z/5Z {0, 1, 2, 3, 4}, f(m) = r 5 (m)

4 4 ALLAN YASHINSKI This argument is typical of how one verifies that a function f : X/ Y is well-defined. One takes arbitrary elements x, x X for which x x, and then one shows that the proposed expressions for f(x) and f(x ) give the same element of Y. In this case we would say that the expression f(x) depends only on the equivalence class of x, not on the representative x itself. To prove the theorem, we must take a slight digression into the foundations of what a function actually is. 3. The Definition of a Function We have been dealing with functions for quite some time now, but we never actually gave them a proper definition. According to the text: A function f from X to Y, written f : X Y, is a rule that pairs an element x X with an element y Y, written f(x) = y, such that the following property holds. ( x X)(!y Y )f(x) = y. The central issue here is that the term rule is undefined. This makes for an ambiguous definition. This is really unacceptable in mathematics. Everything must always have a precise, unambiguous definition. This deficiency (hopefully) did not impede our understanding of functions. Instead, it is making it quite awkward to prove the above theorem. In order to prove that something is a function, we must be clear about what a function is. Modern mathematics is built upon the foundations of set theory. What this means, is that everything in mathematics is really a set. This includes numbers, ordered pairs, relations (as we saw in class), and functions. We often don t think of these objects as sets, but they must be defined as sets so that we can be clear about what they are when we need to be. Here is the definition of a function: Definition. A function f from a set X to a set Y, is a subset of X Y with the following property: ( x X)(!y Y ) (x, y) f. The set X is the domain of f and the set Y is the codomain of f. For each x X, the unique value y Y such that (x, y) f is denoted by f(x) and is called the image of x. Notice that f is a set. By definition, the statement (x, y) f is equivalent to the statement f(x) = y. So whenever we say f(x) = y, we really mean (x, y) f. The collection of ordered pairs in f completely encode the rule of f. On your exam, the set f was called the graph of f. Let us now define function composition. Definition. Suppose f X Y and g Y Z are functions. The composition of g with f is the set g f = { (x, z) X Z : ( y Y ) (x, y) f (y, z) g }. Exercise D. Prove that g f, as defined above, is a function. (Hint: for each x X, prove existence and uniqueness of z Z for which (x, z) g f separately. To prove uniqueness, suppose (x, z 1 ), (x, z 2 ) g f, and show that z 1 = z 2.)

5 EQUIVALENCE RELATIONS AND WELL-DEFINEDNESS 5 We can translate the definitions of injectivity and surjectivity in terms of the set f. Definition. Let f X Y be a function. (i) We say f is injective if ( y Y )( x 1, x 2 X) ( (x 1, y) f (x 2, y) f (x 1 = x 2 ) ). (ii) We say f is surjective if ( y Y )( x X) (x, y) f. Exercise E. Suppose that f X Y is a function. Define f 1 Y X by f 1 = {(y, x) Y X : (x, y) f}. Prove that if f is both injective and surjective, then f 1 is a function. Now we shall prove Theorem 1. Proof. (Theorem 1) Suppose is an equivalence relation on X and g : X Y is a function such that Define f (X/ ) Y by g(x) = g(x ) whenever x x. f = {( x, g(x) ) : x X }. We shall prove that f is a function. Notice that every element of X/ is of the form x for some x X. So let x X/ be an arbitrary equivalence class, one of whose elements is x X. Let y = g(x). Then ( x, y ) = ( x, g(x) ) f. This shows that for every x X/, there is y Y such that ( x, y ) f. For the uniqueness assertion, suppose that ( x, g(x) ) f and ( x, g(x ) ) f are such that x = x. Then x x, which implies x x by definition of equivalence class. So g(x) = g(x ), by the assumed properties of g. We conclude that for every x X/, there is a unique y Y such that ( x, y ) f. By definition, f is a function. From now on, we shall no longer explicitly think of a function f : X Y as a subset of X Y, unless we need to. In fact, other people may look strangely at you if you say things like (x, y) f, rather than f(x) = y (remember, they are equivalent statements, by definition.) So now we can go about our lives treating functions as we used to, secretly knowing that we ve built a rigorous foundation of what a function really is. Example 4. Another situation in which we have been a little lax is in defining ordered pairs properly (and consequently Cartesian products). In light of the definition of a function, this is a loose end we should tie up. Everything must be a set, so we need to encode an ordered pair (x, y) into some set. Since sets are unordered, we cannot use {x, y} to represent (x, y), because {x, y} = {y, x} would then represent (y, x) as well. This is a problem because we want to distinguish between (x, y) and (y, x). Someone clever (Kazimierz Kuratowski) came up with

6 6 ALLAN YASHINSKI the following definition. Given x X and y Y, define the ordered pair (x, y) to be the set (x, y) = { {x}, {x, y} }. Notice that (x, y) P(P(X Y )). This turns out to be a good definition, because it satisfies what we want to be true about ordered pairs, namely that (x 1, y 1 ) = (x 2, y 2 ) if and only if (x 1 = x 2 ) (y 1 = y 2 ). Notice here that (x 1, y 1 ) = (x 2, y 2 ) is an equality of sets. The Cartesian Product of X and Y is defined to be the subset X Y P(P(X Y )) given by X Y = { (x, y) : x X y Y }. Exercise F. Using Kuratowski s definition of an ordered pair, prove that (x 1, y 1 ) = (x 2, y 2 ) if and only if (x 1 = x 2 ) (y 1 = y 2 ). (Hint: The forward ( ) direction is the nontrivial part. You may want to deal with the case where x 1 = y 1 first, and then you can assume x 1 y 1 for the rest of the proof. The reason for doing this is that (x, x) = { {x}, {x, x} } = { {x}, {x} } = { {x} } reduces to something much simpler when the two coordinates are the same.) Similarly to the case of functions, no one ever really explicitly thinks of (x, y) as the set { {x}, {x, y} }. For example, people will look strangely at you if you write things like {x} (x, y) or { {x, y} } (x, y), even if they are technically true. What s most important here is that we ve rigorously defined the object (x, y) in such a way that (x 1, y 1 ) = (x 2, y 2 ) if and only if (x 1 = x 2 ) (y 1 = y 2 ). Now, we can treat an ordered pair as a black box that satisfies the above property. If you think about it, that s what we ve been doing all along. 4. A Canonical Bijection Given a function f : X Y, let f be the relation on X defined by x f x if and only if f(x) = f(x ). Exercise G. Prove that f is an equivalence relation on X. Theorem 2. If f : X Y is a surjection, then there is a bijection f : X/ f Y, defined by f ( x ) = f(x). Proof. By definition of f, we know that f(x) = f(x ) whenever x f x. So by Theorem 1, the function f : X/ f Y, ) f( x = f(x)

7 EQUIVALENCE RELATIONS AND WELL-DEFINEDNESS 7 is well-defined. We shall prove f is a bijection. To see it is surjective, let y Y be arbitrary. Since f is surjective, there is some x X for which f(x) = y. Thus, the element x X/ f satisfies f ( x ) = f(x) = y. Thus, f is surjective. To see it is injective, suppose x, x X/ f are arbitrary classes for which f ( x ) = f ( x ). Then f(x) = f(x ) by definition of f. It follows that x f x by definition of f. As f is an equivalence relation, this implies x = x. Thus, f is surjective. As an example of this theorem, consider for n N the surjection r n : Z {0, 1, 2,..., n 1} which associates to an integer its remainder after division by n. Notice that the equivalence relation rn on Z is exactly congruence modulo n. By Theorem 2, the map ( ) r n : Z/nZ {0, 1, 2,..., n 1}, r n m = rn (m) is a bijection. We conclude that Z/nZ contains exactly n distinct equivalence classes, and more explicitly, Z/nZ = { 0, 1, 2,..., n 1 }. 5. Modular Arithmetic The familiar arithmetical operations of addition and multiplication can actually be carried out on the elements of Z/nZ = { 0, 1, 2,..., n 1 } Addition. Notice that the operation of addition of integers can be thought of as a function + : Z Z Z. The element +(a, b) Z is usually written as a + b. To define addition on Z/nZ, we will define a function + : Z/nZ Z/nZ Z/nZ. We shall write a + b to mean the element + ( a, b ) of Z/nZ. Definition. Define + : Z/nZ Z/nZ Z/nZ by a + b = a + b. Notice that the a + b on the right hand side is the sum of two integers, which is already defined. The sum a + b is defined the be the equivalence class of the sum a + b. Here, we run into the potential problem that this operation is not welldefined. The sum a + b is actually defined in terms of the representatives of the equivalence classes. We must make sure that if we choose different representatives, we still get the same result. Here s how a typical proof of well-definedness of this operation looks: Suppose a c mod n and b d mod n. Then a = c + nk, b = d + nl,

10 10 ALLAN YASHINSKI Example 8. In Z/5Z, the following are true: 1 1 = 1, 2 3 = 1, 4 4 = 1. Thus, 3 is the inverse of 2 (and vice-versa), 4 is its own inverse, and 1 is its own inverse. Example 9. In Z/4Z, the element 2 has no multiplicative inverse. We can prove this by exhaustion (checking its product with all possibilities): 2 0 = 0, 2 1 = 2, 2 2 = 0, 2 3 = 2. Thus, there is no element a such that 2 a = 1. Exercise H. (a) It is a fact that in Z/7Z = { 0, 1, 2, 3, 4, 5, 6 }, every element except for 0 has a multiplicative inverse. Find the inverse of each nonzero class. (b) Determine which elements of Z/9Z = { 0, 1, 2, 3, 4, 5, 6, 7, 8 } have a multiplicative inverse, and find the inverses. A commutative ring for which each nonzero element has a multiplicative inverse is called a field. Once again, this is more abstract algebra terminology that you don t have to know for this class. It is a fact, which we shall neither prove nor use, that Z/nZ is a field if and only if n is prime. 6. Building the Rational Numbers As discussed above, everything in mathematics must be defined, typically as a set, and this includes numbers. Defining our number systems N, Z, Q, R, and C is an issue we have largely avoided, and is one which may be more complicated than you would think. Notice that N Z Q R C. This chain of subset inclusions also represents how one constructs these number systems: Z is defined in terms of N. Q is defined in terms of Z. R is defined in terms of Q. C is defined in terms of R. This begs the question: How does one define N? The answer is that N must be defined using sets, and sets alone. We shall not explore the construction of N, but it is a fact that once we have defined the set N, we can build the other number systems as indicated above. Each of the four constructions differs from the others. Some are simpler than others: Defining the complex numbers C in terms of the reals R is one of the simplest, whereas defining R in terms of Q is much more difficult. Here, we shall explore only one of them: constructing the set Q of rational numbers from the set Z of integers. Here is the main idea in defining Q. We think of a rational number as a quotient a b of two integers a and b. We could attempt to represent this quotient as an ordered

11 EQUIVALENCE RELATIONS AND WELL-DEFINEDNESS 11 pair (a, b) Z Z (remember we are supposed to be using Z to define Q.) The central issue is this: several pairs of integers can be used to represent the same rational number. For example, we all know that 1 2 = 2 4 = 5 10 = However, none of the ordered pairs (1, 2), (2, 4), (5, 10), (150, 300) are equal, as elements on Z Z. We shall want to think of these ordered pairs as being the same. The mathematical solution to this problem is to put an equivalence relation on the set of ordered pairs of integers in such a way that the above pairs are equivalent. This is the idea that will guide the construction. Not every ordered pair of integers should represent a fraction, because we want to prohibit the situation in which the denominator is 0. Thus, the set we shall work with will be Z ( Z \ {0} ). An element (a, b) Z ( Z \ {0} ) will ultimately represent the number a b. We need to figure out when two such pairs (a, b) and (c, d) represent the same fraction. We ve known since grade school that the statement a b = c if and only if ad = bc d needs to be true. Teachers usually call this cross-multiplying. This motivates the following definition. Definition. Define a relation on Z ( Z \ {0} ) so that (a, b) (c, d) whenever ad = bc. Exercise I. Prove that is an equivalence relation on Z ( Z \ {0} ). (Hint: To prove transitivity, assume (a, b) (c, d) and (c, d) (e, f). Then show that this implies that d(af be) = 0. Then use the fact that if a product of integers is zero, then one of the factors must be zero. Definition. The set Q of rational numbers is defined the be the set of equivalence classes of elements of the set Z ( Z \ {0} ) with respect to the relation. Given a, b Z with b 0, se shall use the notation a b to mean (a, b) Q. That is, a b = (a, b), is a definition, where we are defining the left hand side to be the right hand side. Exercise J. Prove that 1 2 = 2 4 and 2 5 = Exercise K. Define a function f : Z Q by f(n) = (n, 1). Prove that this function is injective. (Be careful about what it means when two elements of Q are equal.) This allows us to identify Z as a subset of Q, namely its image f(z) Q, under the function f.

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